Question

Solve each system of inequalities by graphing.

$\begin{array}{c}y\ge x-3\\ y\ge -x+1\end{array}$

Short answer

The solution of the inequalities is$-y+1\le x\le y+3$.

Step-by-step solution

Step-1 – Concept of solution of linear inequalities

The solution of linear inequalities can be obtained by changing the inequalities into equations and solving the linear equations to obtain a graph. Then the common shaded region is a solution of the inequalities.

Step-2 – Concept of shading the region of the inequalities

The shaded region obtained by choosing a point, if the point satisfies the inequality the region is along the point, if not satisfies the inequalities, then the shaded region is opposite to the point.

Step-3 – Solving the inequalities

The given inequalities are-:

$\begin{array}{c}y\ge x-3\\ y\ge -x+1\end{array}$

The linear equation of the inequalities is-:

$\begin{array}{c}y=x-3\\ y=-x+1\end{array}$

The point which satisfies the equation $y=x-3$ are $(0,-3)$and$(3,0)$.

The point which satisfies the equation $y=-x+1$ are $(0,1)$ and$(1,0)$.

Step-4 – Evaluating the shading region

We choose $(0,0)$then the point $(0,0)$ satisfies the inequality $y\ge x-3$ but it does not satisfy the inequality $y\ge -x+1$.

Step-5 – Plotting the graph

So, the graph of the inequality is

The common region is$-y+1\le x\le y+3$